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A236400 Primes p=prime(k) such that min{r_p, p-r_p} <= 2, where r_p = A100612(k).

%I A236400 #20 Jan 03 2022 16:59:21
%S A236400 2,3,5,7,11,23,31,67,227,373,10331,274453
%N A236400 Primes p=prime(k) such that min{r_p, p-r_p} <= 2, where r_p = A100612(k).
%C A236400 No further terms < 5*10^6. - _Michael S. Branicky_, Jan 03 2022
%H A236400 Romeo Mestrovic, <a href="http://arxiv.org/abs/1312.7037">Variations of Kurepa's left factorial hypothesis</a>, arXiv preprint arXiv:1312.7037 [math.NT], 2013-2014.
%H A236400 Miodrag Zivkovic, <a href="https://doi.org/10.1090/S0025-5718-99-00990-4">The number of primes sum_{i=1..n} (-1)^(n-i)*i! is finite</a>, Math. Comp. 68 (1999), pp. 403-409.
%p A236400 A100612 := proc(n)
%p A236400     local p,lf,kf,k ;
%p A236400     p := ithprime(n) ;
%p A236400     lf := 1 ;
%p A236400     kf := 1 ;
%p A236400     for k from 1 to p-1 do
%p A236400         kf := modp(kf*k,p) ;
%p A236400         lf := lf+modp(kf,p) ;
%p A236400     end do:
%p A236400     lf mod p ;
%p A236400 end proc:
%p A236400 for n from 1 do
%p A236400     p := ithprime(n) ;
%p A236400     rp := A100612(n) ;
%p A236400     prp := p-rp ;
%p A236400     if min(rp,prp) <= 2 then
%p A236400         print(p) ;
%p A236400     end if;
%p A236400 end do: # _R. J. Mathar_, Feb 17 2014
%t A236400 A100612[n_] := Module[{p = Prime[n], lf = 1, kf = 1, k}, For[k = 1, k <= p - 1, k++, kf = Mod[kf*k, p]; lf = lf + Mod[kf, p]]; Mod[lf, p]];
%t A236400 Reap[For[n = 1, n < 40000, n++, p = Prime[n]; rp = A100612[n]; If[Min[rp, p - rp] <= 2, Print[p]; Sow[p]]]][[2, 1]] (* _Jean-François Alcover_, Dec 05 2017, after _R. J. Mathar_ *)
%o A236400 (Python)
%o A236400 from sympy import isprime
%o A236400 def afind(limit):
%o A236400     f = 1 # (p-1)!
%o A236400     s = 2 # sum(0! + 1! + ... + (p-1)!)
%o A236400     for p in range(2, limit+1):
%o A236400         if isprime(p):
%o A236400             r_p = s%p
%o A236400             if min(r_p, p-r_p) <= 2:
%o A236400                 print(p, end=", ")
%o A236400         s += f*p
%o A236400         f *= p
%o A236400 afind(11000) # _Michael S. Branicky_, Jan 03 2022
%Y A236400 Cf. A003422, A236399, A100612.
%K A236400 nonn,more
%O A236400 1,1
%A A236400 _N. J. A. Sloane_, Jan 29 2014
%E A236400 a(12) from _Jean-François Alcover_, Dec 05 2017